First of all I examined by ANOVA if there is a weekend effect. I chose $\alpha=0.05$. For the $F$-statstic I got $F=0.1$ and a p-value of $0.76$. So the null hypothesis that $\theta_1=\theta_2$ can not be fefused and so there is no weekend effect.

(a) The estimated coefficients are
$$
\hat{\theta}_1=-1.0019,~~~\hat{\theta}_2=-0.1032,~~~\hat{\theta}_3=-0.0025,
$$
where the first coefficient is the intercept, the second coefficient belongs to $X_1$ and the thirds one to $X_2$.

Now I do not know what I should comment here? What could be meant? What is to be commented?

(b)

The global $F$-test says that the hypothesis $H_0: \theta_2=\theta_3=0$ can be refused. So the regressors play a role for explaining the $SO_2$-concentration.

(c)

For $\theta_2$ the confidence interval is
$$
C_2^{0.05}=(-0.156863,-0.049537),
$$
so because $0\notin C_{2}^{0.05}$, the hypothesis $H_0^2: \theta_2=0$ can be refused.

For $\theta_3$ the confidence intervall is given by
$$
C_3^{0.05}=(-0.437834,0.432834),
$$
so because of $0\in C_3^{0.05}$, the hypothesis $H_0^3: \theta_3=0$ cannot be refused, so the covariable $X_3$ does not have an influence and can be removed from the model.

I now made the simple linear regression, again using R, getting now
$$
\hat{\theta}_1=-1.0020,~~~\hat{\theta}_2=-0.1033.
$$

What I see is, that the coefficients are nearly the same compared to those of the multiple regression.

But I do not know how to compare the simple linear regression and the multiple linear regression graphically. Of course, I can plot the simple linear regression:

but I do not know how to plot the multiple regression in order to compare it then with the simple linear regression. Can you help me with that?

So to sum it up:

Concerning (a) I do not know what is to be commented and concerning (c) I do not know how I can compare the multiple and the simple regression graphically.

1 Answer
1

When I hear the question, my concrete physical reality self kicks in and asks it slightly differently. I hear "does the answer make sense" and "what do the numbers mean". What does it imply if there is a weekend effect?

When I want to compare - I would look at error histograms. I would compute error for the simple regression, and look at the distribution plot. (link) I would then compute error for the multiple regression and look at the distribution plot. Using these I could look at typical error, maximum error between the fits.

I'm wondering if the negative value is related to the idea of pH. If there is a logarithm of a concentration between 0 and 100% then a negative prefix would be required for anything but pure sulfur dioxide. For all $x<1$ the value of $log_{10}\left(x\right)$ is negatively valued. When I check this (link) then I find that 800 micrograms per cubic meter is an alert-level of sulfur dioxide. If I compute $ 10^{-y}$ I can get on the order of ~800 units for some of the values. The measurement is one per this, so around 1 part per 800. I think that they compute how many micrograms per cubic meter, then take the $log_{10}$ of that.